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Fourier Transform for Mean Periodic Functions

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Fourier Transform for Mean Periodic Functions Annales Univ. Sci. Budapest., Sect. Comp. 35 (2011) 267–283 FOURIER TRANSFORM FOR MEAN PERIODIC FUNCTIONS L´aszl´oSz´ekelyhidi (Debrecen, Hungary) Dedicated to the 60th birthday of Professor Antal J´arai Abstract. Mean periodic functions are natural generalizations of periodic functions. There are different transforms - like Fourier transforms - defined for these types of functions. In this note we introduce some transforms and compare them with the usual Fourier transform. 1. Introduction In this paper C(R) denotes the locally convex topological vector space of all continuous complex valued functions on the reals, equipped with the linear operations and the topology of uniform convergence on compact sets. Any closed translation invariant subspace of C(R)iscalledavariety. The smallest variety containing a given f in C(R)iscalledthevariety generated by f and it is denoted by τ(f). If this is different from C(R), then f is called mean periodic. In other words, a function f in C(R) is mean periodic if and only if there exists a nonzero continuous linear functional μ on C(R) such that (1) f ∗ μ =0 2000 Mathematics Subject Classification: 42A16, 42A38. Key words and phrases: Mean periodic function, Fourier transform, Carleman transform. The research was supported by the Hungarian National Foundation for Scientific Research (OTKA), Grant No. NK-68040. 268 L. Sz´ekelyhidi holds. In this case sometimes we say that f is mean periodic with respect to μ. As any continuous linear functional on C(R) can be identified with a compactly supported complex Borel measure on R, equation (1) has the form (2) f(x − y) dμ(y)=0 for each x in R. The dual of C(R) will be denoted by MC (R). As the convolu- tion of two nonzero compactly supported complex Borel measures is a nonzero compactly supported Borel measure as well, all mean periodic functions form a linear subspace in C(R). We equip this space with the following topology. For each nonzero μ from the dual of C(R)letV (μ) denote the solution space of (1). Clearly, V (μ) is a variety and the set of all mean periodic functions is equal to the union of all these varieties. We equip this union with the inductive limit of the topologies of the varieties V (μ) for all nonzero μ from the dual of C(R). The locally convex topological vector space obtained in this way will be denoted by MP(R), the space of mean periodic functions. An important class of mean periodic functions is formed by the exponential polynomials. We call a function of the form (3) ϕ(x)=p(x) eλx an exponential monomial,wherep is a complex polynomial and λ is a complex number. If p ≡ 1, then the corresponding exponential monomial x → eλx is called an exponential. Exponential monomials of the form k λx (4) ϕk(x)=x e with some natural number k and complex number λ, are called special expo- nential monomials. Linear combinations of exponential monomials are called exponential poly- nomials. To see that the special exponential monomial in (3) is mean periodic one considers the measure λ k+1 (5) μk =(e δ1 − δ0) , where δy is the Dirac–measure concentrated at the number y for each real y, and the k + 1-th power is meant in convolution-sense. It is easy to see that ϕk ∗ μk =0 holds. Sometimes we write 1 for δ0. Exponential polynomials are typical mean periodic functions in the sense that any mean periodic function f in V (μ) is the uniform limit on compact sets of a sequence of linear combinations of exponential monomials, which belong to V (μ), too. More precisely, the following theorem holds (see [9]). Fourier transform for mean periodic functions 269 Theorem 1 (L. Schwartz, 1947). In any variety of C(R) the linear hull of all exponential monomials is dense. A similar theorem in C(Rn) fails to hold for n ≥ 2 as it has been shown in [4] by D. I. Gurevich. Moreover, he gave examples for nonzero varieties in C(R2) which do not contain nonzero exponential monomials at all. However, as it has been shown by L. Ehrenpreis in [1], Theorem 1 can be extended to varieties of the form V (μ)inC(Rn) for any positive integer n. Another important result in [9] is the following (Th´eor`eme 7, on p. 881.): Theorem 2. In any proper variety of C(R) no special exponential monomial is contained in the closed linear hull of all other special exponential monomials in the variety. In other words, if a variety V = {0} in C(R) is given, then for each special exponential monomial ϕ0 in V there exists a measure μ in MC (R) such that μ(ϕ0)=1andμ(ϕ) = 0 for each special exponential monomial ϕ = ϕ0 in V . 2. A mean operator for mean periodic functions Based on Theorems 1 and 2 by L. Schwartz we introduced a mean operator on the space MP(R) in the following way (see also [10], pp. 64–65.). For each x, y in R and f in C(R)let τyf(x)=f(x + y) , and call τyf the translate of f by y. The continuous linear operator τy on C(R) is called translation operator. The operator τ0 will be denoted by 1. Clearly, the continuous function f is a polynomial of degree at most k if and only if k+1 (6) (τy − 1) f(x)=0 holds for each x, y in R and for k =0, 1,.... The set P(R) of all polynomials is a subspace of MP(R), which we equip with the topology inherited from MP(R). Theorem 3. The subspace P(R) is closed in MP(R). Proof. First we show that the set of the degrees of all polynomials in any proper variety is bounded from above. By the Taylor–formula it follows that 270 L. Sz´ekelyhidi the derivative of a polynomial is a linear combination of its translates, hence if a polynomial belongs to a variety then all of its derivatives belong to the same variety, too. Therefore, if the set of the degrees of all polynomials in a proper variety is not bounded from above, then all polynomials belong to this variety. But, in this case, by the Stone–Weierstrass–theorem, all continuous functions must belong to the variety, hence it cannot be proper. Suppose now that (pi)i∈I is a net of polynomials which converges in MP(R) to the continuous function f. By the definition of the inductive limit topology there exists a nonzero μ in Mc(R) such that pi belongs to V (μ)foreachi in I. By our previous consideration, for the degrees we have deg pi ≤ k for some positive integer k. By (6) this means that k+1 (τy − 1) pi(x)=0 holds for each x, y in R. This implies that the same holds for f,hencef is a polynomial of degree at most k, too. The theorem is proved. Theorem 4. There exists a unique continuous linear operator M : MP(R) →P(R) satisfying the properties 1) M(τyf)=τy M(f), 2) M(p)=p for each f in MP(R), p in P(R) and y in R. Proof. First we prove uniqueness. By Theorem 1, it is enough to show that the properties of M determine M on the set of all special exponential monomials. Let m = 1 be any nonzero continuous complex exponential. Then we have M(m)=M m(−y)τym = m(−y)M(τym)=m(−y)τyM(m), which implies that either M(m)=0orm is a polynomial. Hence M(m)=0. Suppose that we have proved for j =0, 1,...,k− 1that M xjm(x) =0 for any continuous complex exponential m =1.Thenwehave $ k % k M (x + y)km(x + y) = M xjyk−jm(x)m(y) = j j=0 Fourier transform for mean periodic functions 271 k k = yk−jm(y)M xjm(x) = m(y)M xkm(x) , j j=0 which implies, as above, that M xkm(x) = 0. This proves the uniqueness. In order to prove existence, first we notice, that, by Theorem 2, for any nonzero μ in Mc(R), the exponential 1 is not contained in the closed linear subspace of C(R) spanned by all special exponential monomials in V (μ) different from 1. This implies the existence of a measure μ0 in Mc(R) such that μ0(1) = = 1, further μ0(ϕ) = 0 for any special exponential monomial ϕ =1in V (μ). k From this fact it follows, that x m(x) ∗ μ0 =0foreachpositiveintegerk k k and exponential m =1in V (μ), further x ∗ μ0 = x . This shows, that ϕ ∗ μ0 is a polynomial in V (μ) for any exponential polynomial ϕ in V (μ). On the other hand, as in the proof of Theorem 3, it follows that if a polynomial of degree n belongs to V (μ), then all the functions 1,x,x2,...,xn also belong to V (μ). Hence, all polynomials in V (μ) have a degree smaller than some fixed positive integer. Now, if f is arbitrary in V (μ), then by Theorem 1, there exist exponential polynomials ϕi in V (μ) such that f =limϕi.Thenwehave f ∗ μ0 = lim(ϕi ∗ μ0), hence also f ∗ μ0 is a polynomial. Suppose now, that f belongsalsotosomeV (ν) with some nonzero ν.Then f ∗ μ0 also belongs to V (ν), and it is a polynomial. Hence we have f ∗ μ0 = = f ∗ μ0 ∗ ν0.
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